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Numerical Discrete-Time Implementation of Continuous-Time Linear-Quadratic Model Predictive Control

Zhanhao Zhang, Anders Hilmar Damm Christensen, Steen Hørsholt, John Bagterp Jørgensen

TL;DR

The paper addresses implementing a continuous-time linear-quadratic model predictive control (CT-LMPC) for systems with time delays and stochastic disturbances by discretizing a CT LQ-OCP into a discrete-time quadratic program. It employs a Noise-Separation (NS) state-space representation to split deterministic and stochastic dynamics, performs Kalman-filter-based estimation for the stochastic part, and derives discrete cost terms for reference tracking, input regularization, input rate ROM, and soft output constraints, culminating in a unified DT-QP formulation. Numerical results on SISO and MIMO (cement-mill style) plants show CT-LMPC can achieve performance comparable to standard DT-LMPC at moderate sampling times but outperforms it at larger sampling times, with the gap widening as the controller rate slows. The work provides a practical framework for designing NMPC-like controllers from a CT-LQ-OCP, offering improved robustness to delays and disturbances while maintaining tractable optimization.

Abstract

This study presents the design, discretization and implementation of the continuous-time linear-quadratic model predictive control (CT-LMPC). The control model of the CT-LMPC is parameterized as transfer functions with time delays, and they are separated into deterministic and stochastic parts for relevant control and filtering algorithms. We formulate time-delay, finite-horizon CT linear-quadratic optimal control problems (LQ-OCPs) for the CT-LMPC. By assuming piece-wise constant inputs and constraints, we present the numerical discretization of the proposed LQ-OCPs and show how to convert the discrete-time (DT) equivalent into a standard quadratic program. The performance of the CT-LMPC is compared with the conventional DT-LMPC algorithm. Our numerical experiments show that, under fixed tunning parameters, the CT-LMPC shows better closed-loop performance as the sampling time increases than the conventional DT-LMPC.

Numerical Discrete-Time Implementation of Continuous-Time Linear-Quadratic Model Predictive Control

TL;DR

The paper addresses implementing a continuous-time linear-quadratic model predictive control (CT-LMPC) for systems with time delays and stochastic disturbances by discretizing a CT LQ-OCP into a discrete-time quadratic program. It employs a Noise-Separation (NS) state-space representation to split deterministic and stochastic dynamics, performs Kalman-filter-based estimation for the stochastic part, and derives discrete cost terms for reference tracking, input regularization, input rate ROM, and soft output constraints, culminating in a unified DT-QP formulation. Numerical results on SISO and MIMO (cement-mill style) plants show CT-LMPC can achieve performance comparable to standard DT-LMPC at moderate sampling times but outperforms it at larger sampling times, with the gap widening as the controller rate slows. The work provides a practical framework for designing NMPC-like controllers from a CT-LQ-OCP, offering improved robustness to delays and disturbances while maintaining tractable optimization.

Abstract

This study presents the design, discretization and implementation of the continuous-time linear-quadratic model predictive control (CT-LMPC). The control model of the CT-LMPC is parameterized as transfer functions with time delays, and they are separated into deterministic and stochastic parts for relevant control and filtering algorithms. We formulate time-delay, finite-horizon CT linear-quadratic optimal control problems (LQ-OCPs) for the CT-LMPC. By assuming piece-wise constant inputs and constraints, we present the numerical discretization of the proposed LQ-OCPs and show how to convert the discrete-time (DT) equivalent into a standard quadratic program. The performance of the CT-LMPC is compared with the conventional DT-LMPC algorithm. Our numerical experiments show that, under fixed tunning parameters, the CT-LMPC shows better closed-loop performance as the sampling time increases than the conventional DT-LMPC.
Paper Structure (11 sections, 48 equations, 2 figures, 2 algorithms)

This paper contains 11 sections, 48 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Discretization of Model Predictive Control
  3. Discretization of control model
  4. Reference tracking and input regularization objectives
  5. Input ROM and economics objectives
  6. Soft output constraint penalty
  7. Design and implementation of CT-LMPC
  8. Numerical Experiments
  9. Closed-loop simulation - a SISO example
  10. Closed loop simulation - a MIMO example
  11. Conclusions

Figures (2)

  • Figure 1: SISO example closed-loop simulations with different controller sampling times $T^c_{s} = 5, 15, 25$ [s]. The blue curves indicate the results obtained from DT-LMPC and the black curves are the results of CT-LMPC.
  • Figure 2: The closed-loop simulations of a simulated cement mill system with DT-LMPC (blue curves) and CT-LMPC (black curves) implementations. The simulator and controller sampling times are $T_s$ = 1 [min] and $T_s^c$ = 2 [min]